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Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that 

(i) A × (B ∩ C) = (A × B) ∩ (A × C) 

(ii) A × C is a subset of B × D

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(i) To verify: A × (B ∩ C) = (A × B) ∩ (A × C) We have B ∩ C = {1, 2, 3, 4} ∩ {5, 6} = Φ 

∴ L.H.S. = A × (B ∩ C) = A × Φ = Φ

 A × B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}

 A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}  

∴ R.H.S. = (A × B) ∩ (A × C) = Φ 

∴ L.H.S. = R.H.S Hence, A × (B ∩ C) = (A × B) ∩ (A × C)

 (ii) To verify: A × C is a subset of B × D

 A × C = {(1, 5), (1, 6), (2, 5), (2, 6)} 

A × D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)} 

We can observe that all the elements of set A × C are the elements of set B × D. Therefore, A × C is a subset of B × D. 

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